A218585 -id:A218585 - cp's OEIS frontend

A218754 Number of ways to write n=p+q(3+(-1)^n)/2 with q<=n/2 and p, q, p^2+3pq+q^2 all prime.

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 2, 1, 3, 1, 3, 1, 1, 2, 1, 0, 3, 3, 2, 3, 3, 0, 3, 0, 3, 2, 1, 1, 4, 1, 2, 2, 1, 2, 0, 2, 2, 2, 3, 0, 4, 1, 1, 2, 0, 1, 2, 3, 5, 0, 2, 1, 3, 4, 1, 1, 2, 2, 6, 2, 2, 4, 1, 2, 3, 2, 3, 3, 3, 2, 4, 1, 2, 5, 0, 3, 4, 2, 3, 4, 3, 1, 4, 3

Offset: 1

Zhi-Wei Sun, Nov 04 2012

nonn

Conjecture: a(n)>0 for all n>=1188.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 23 2023
This conjecture is stronger than both Goldbach's conjecture and Lemoine's conjecture.
Zhi-Wei Sun also made the following conjecture: Given any positive odd integer d, there is a prime p(d) such that for any prime p>p(d) there is a prime q
Conjecture verified for d up to 100 and p up to 10^7. - Mauro Fiorentini, Sep 23 2023

			For n=72 we have a(72)=1 since the only primes p and q with p+q=72, q<=36 and p^2+3pq+q^2 prime are p=67 and q=5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002372, A046927, A218585, A218654, A218656.

Cf. A000034 = 1,2,1,2,... = (3-(-1)^n)/2. (Note: Offset shifted w.r.t. use in the definition of this sequence.) - M. F. Hasler, Nov 05 2012

a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[n-q(3-(-1)^n)/2]&&PrimeQ[q^2+3q(n-q(3-(-1)^n)/2)+(n-q(3-(-1)^n)/2)^2]==True,1,0],{q,1,n/2}]
Do[Print[n," ",a[n]],{n,1,20000}]

A218654 Number of ways to write n as x+y with 0

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 2, 3, 2, 6, 3, 4, 3, 6, 2, 6, 3, 4, 3, 8, 3, 8, 2, 5, 5, 8, 4, 8, 6, 5, 4, 8, 2, 10, 6, 6, 3, 11, 4, 9, 6, 9, 7, 10, 4, 14, 6, 9, 3, 11, 4, 12, 7, 9, 10, 10, 4, 11, 5, 10, 9, 15, 4, 15, 9, 9, 8, 14, 6, 12, 8, 9, 8, 18, 4, 17, 11, 9, 11, 20, 5, 14, 10, 13, 7, 16, 9, 17, 6, 16, 10

Offset: 1

Zhi-Wei Sun, Nov 03 2012

nonn

Conjecture: a(n)>0 for all n=2,3,4,...
It is known that any prime p = 1 or -1 (mod 5) can be written uniquely in the form x(p)^2+3x(p)y(p)+y(p)^2 with x(p)>y(p)>0.
Zhi-Wei Sun also conjectured that
   (sum_{p     /(sum_{phas the limit 1+sqrt(5) as N tends to the infinity.
These conjectures are similar to the ones mentioned in the comments in A218585.

			For n=12 we have a(12)=1 since x^2+3x(12-x)+(12-x)^2 with 0<x<=6 is prime only when x=5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Cf. A038872, A218585.

a[n_]:=a[n]=Sum[If[PrimeQ[x^2+3x(n-x)+(n-x)^2]==True,1,0],{x,1,n/2}]; Do[Print[n," ",a[n]],{n,1,20000}]
Table[Count[IntegerPartitions[n,{2}],?(PrimeQ[#[[1]]^2+3Times@@#+ #[[2]]^2]&)], {n,110}] (* _Harvey P. Dale, Feb 28 2018 *)

A218654(n)=sum(x=1,n\2,isprime(x^2+(2*x+n)*(n-x)))  \\ - M. F. Hasler, Nov 05 2012

A218825 Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 1, 3, 3, 1, 2, 2, 1, 2, 3, 1, 2, 3, 1, 2, 2, 1, 3, 1, 1, 3, 3, 4, 3, 1, 2, 5, 3, 1, 3, 2, 4, 3, 3, 1, 7, 4, 1, 5, 3, 5, 8, 4, 3, 4, 3, 3, 5, 4, 4, 3, 2, 3, 5, 3, 5, 7, 3, 2, 9, 4, 4, 6, 3, 3, 8, 6, 1, 4, 5, 2, 7, 1, 4, 2, 4, 5, 5, 2, 4, 4, 3, 2, 5, 4, 5, 6, 4, 1

Offset: 1

Zhi-Wei Sun, Nov 07 2012

nonn

Conjecture: a(n)>0 for all n>8.
This conjecture is stronger than Lemoine's conjecture. It has been verified for n up to 10^8.
Conjecture verified for 2n-1 up to 10^9. - Mauro Fiorentini, Jul 20 2023
Zhi-Wei Sun also made the following general conjecture: For any positive integer n, the set E(n) of positive odd integers not of the form p+2q with p, q, p^2+4(2^n-1)q^2 all prime, is finite. In particular, if we let M(n) denote the maximal element of E(n), then M(1)=3449, M(2)=1711, E(3)={1,3,5,7,31,73}, E(4)={1,3,5,7,9,11,13,15},
  M(5)=6227, M(6)=1051, M(7)=2239, M(8)=2599, M(9)=7723,
  M(10)=781, M(11)=1163, M(12)=587, M(13)=11443,
  M(14)=2279, M(15)=157, M(16)=587, M(17)=32041,
  M(18)=1051, M(19)=2083, M(20)=4681.
Conjecture verified for 2n-1 up to 10^9 for n <= 4 and up to 10^6 for n <= 20. - Mauro Fiorentini, Jul 20 2023
Zhi-Wei Sun also guessed that for any positive even integer d not congruent to 2 modulo 6 there exists a prime p(d) such that for any prime p>p(d) there is a prime q
  p(4)=p(6)=3, p(10)=5, p(12)=3, p(16)=2, p(18)=3,
  p(22)=11, p(24)=17, and p(28)=p(30)=7.

			a(10)=1 since the only primes p and q with p^2+60q^2 prime and p+2q=19 are p=13 and q=3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
Wikipedia, Lemoine's conjecture

Cf. A000040, A046927, A218754, A218585, A218654, A218656, A218797.

a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[2n-1-2q]==True&&PrimeQ[(2n-1-2q)^2+60q^2]==True,1,0],{q,1,n-1}]
Do[Print[n," ",a[n]],{n,1,20000}]

A218825(n)={my(c=0,n21=n*2-1);forprime(q=2,n-1,isprime(n21-2*q) || next; isprime(q^2*60+(n21-2*q)^2) && c++); c}  \\ M. F. Hasler, Nov 07 2012

A218656 Number of ways to write 2n+1 as x+y with 0 < x < y and x^4 + y^4 prime.

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 1, 5, 4, 4, 4, 5, 4, 7, 6, 5, 3, 10, 4, 9, 8, 4, 9, 6, 7, 11, 7, 5, 11, 9, 9, 9, 11, 4, 14, 14, 9, 8, 9, 7, 11, 8, 12, 12, 10, 9, 11, 17, 10, 12, 16, 7, 13, 14, 8, 15, 9, 11, 23, 16, 9, 17, 23, 8, 15, 15, 11, 21, 18, 12, 19, 14, 15, 19, 21, 17, 16, 23, 13, 21, 20, 17, 29

Offset: 1

Zhi-Wei Sun, Nov 04 2012

nonn

Conjecture: a(n) > 0 for all n >= 1.
Thomas Ordowski conjectured on Nov 03 2012 that if x^4 + y^4 in the definition of a(n) is replaced by x^2 + y^2, then a(n) will always be positive.
We also have similar conjectures with x^4 + y^4 replaced by x^8 + y^8 or x^16 + y^16.
All conjectures verified for 2n+1 up to 10^6: no exceptions for x^2 + y^2 and x^4 + y^4; exceptions 2n+1 = 7, 9, 55, 73, 75 and 105 for x^8 + y^8; exceptions 2n+1 = 5 and 9 for x^16 + y^16. - Mauro Fiorentini, Sep 22 2023
Alternate definition: Number of primes of the form k^4 + (2n+1-k)^4, 0 < k <= n. - M. F. Hasler, Nov 05 2012

			For n=7 we have a(7)=1, since x^4 + (15-x)^4 with 0 < x < 8 is prime only when x=4.

Thomas Ordowski, Personal e-mail message, Nov 03 2012.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Cf. A002645, A218585, A218654.

A218656 := n-> add(`if`(isprime(i^4+(2*n+1-i)^4), 1, 0), i=1..n): # Alois P. Heinz, Jul 09 2016

a[n_]:=a[n]=Sum[If[PrimeQ[x^4+(2n+1-x)^4]==True,1,0],{x,1,n}]
Do[Print[n," ",a[n]],{n,1,20000}]

A218586(n)=sum(x=1,n+0*n=2*n+1, isprime(x^4+(n-x)^4))  \\ M. F. Hasler, Nov 05 2012

A219052 Number of ways to write n = p + q(3 - (-1)^n)/2 with q <= n/2 and p, q, p^2 + q^2 - 1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 0, 2, 1, 0, 0, 1, 1, 3, 0, 1, 1, 1, 1, 3, 1, 1, 4, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 4, 0, 0, 3, 0, 1, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 4, 2, 1, 2, 1, 1, 0, 4, 2, 1, 1, 1, 2, 5, 4, 1, 3, 1, 1, 4, 1, 1, 2, 2

Offset: 1

Zhi-Wei Sun, Nov 10 2012

Conjecture: a(n) > 0 for all n > 784.
This conjecture implies Goldbach's conjecture, Lemoine's conjecture, and that there are infinitely many primes of the form p^2 + q^2 - 1 with p and q both prime.
It has been verified for n up to 10^8.
Zhi-Wei Sun also made the following general conjecture: Let d be any odd integer not congruent to 1 modulo 3. Then, all large even numbers can be written as p + q with p, q, p^2 + q^2 + d all prime. If d is also not divisible by 5, then all large odd numbers can be represented as p + 2q with p, q, p^2 + q^2 + d all prime.

			a(12) = 1 since {5, 7} is the only prime pair {p, q} for which  p + q = 12, and p^2 + q^2 - 1 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Cf. A000040, A002375, A046927, A218754, A218585, A218654, A218825, A219023, A219026.

a[n_] := a[n] = Sum[If[PrimeQ[n - (1 + Mod[n, 2])Prime[k]] == True && PrimeQ[Prime[k]^2 + (n - (1 + Mod[n, 2])Prime[k])^2 - 1] == True, 1, 0], {k, 1, PrimePi[n/2]}]; Do[Print[n, " ", a[n]], {n, 1, 20000}]

A219782 Number of ways to write n=x+y (0

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 3, 2, 3, 1, 1, 0, 2, 0, 2, 1, 3, 1, 2, 1, 3, 2, 4, 2, 2, 1, 1, 2, 4, 2, 3, 2, 4, 3, 0, 1, 2, 2, 1, 0, 4, 1, 4, 1, 6, 2, 1, 2, 6, 1, 3, 0, 1, 3, 5, 2, 7, 2, 1, 2, 4, 1, 3, 3, 5, 2, 1, 2, 2, 2, 4, 0, 3, 1, 5, 2, 4, 3, 2, 3, 2, 3, 2, 1, 4, 3, 3, 2, 3, 2, 7, 1, 5, 5

Offset: 1

Zhi-Wei Sun, Nov 27 2012

nonn

Conjecture: a(n)>0 if n is not among 1, 8, 10, 18, 20, 41, 46, 58, 78, 116, 440.
Zhi-Wei Sun also made the following general conjecture:
For any k=0,1,2,4,5,6,... and positive odd integer m, each sufficiently large integer n can be written as x+y (0For example, if n>6 is different from 24 then n can be written as x+y with x,y positive, and xy-n and xy+n both prime; if n>308 then n can be written as x+y with x,y positive, and 3n^2-xy and 3n^2+xy both prime.

			a(9)=2 since 9=1+8=4+5 with 9^2+1*8, 9^2-1*8, 9^2+4*5, 9^2-4*5 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A091182, A218585, A218654.

a[n_]:=a[n]=Sum[If[PrimeQ[n^2-k(n-k)]==True&&PrimeQ[n^2+k(n-k)]==True,1,0],{k,1,n/2}]
Do[Print[n," ",a[n]],{n,1,10000}]

A218771 Primes of the form p^2 + 3pq + q^2 with p and q prime.

Original entry on oeis.org

31, 59, 79, 179, 191, 229, 251, 311, 389, 401, 479, 491, 541, 569, 719, 809, 971, 1019, 1061, 1109, 1151, 1249, 1301, 1409, 1451, 1499, 1619, 1931, 1949, 2111, 2141, 2339, 2591, 2609, 2711, 2801, 2939, 3089, 3371, 3389, 3449, 3881, 4021, 4091, 4211, 4391, 4451, 4679, 5039, 5051

Offset: 1

Zhi-Wei Sun, Nov 05 2012

nonn

It is easy to see that a(n) is congruent to 1 or 9 modulo 10. For each n there is a unique pair of primes p < q such that p^2 + 3pq + q^2 = a(n).

This sequence is of particular interest due to Zhi-Wei Sun's surprising conjecture related to A218754. That conjecture implies that this sequence is infinite.

			a(1)=31 since 2^2 + 3*2*3 + 3^2 = 31 and 2,3,31 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Cf. A038872, A218754, A218585, A218654, A218656.

SQ[n_]:=IntegerQ[Sqrt[n]]
i=0; Do[Do[If[SQ[4Prime[n]+5Prime[k]^2] && PrimeQ[(Sqrt[4Prime[n] + 5Prime[k]^2] - 3Prime[k])/2] == True, i=i+1; Print[i," ", Prime[n]]; Goto[aa]], {k,1,PrimePi[Sqrt[Prime[n]/5]]}];
Label[aa];Continue,{n,1,1000000}]

list(lim)=my(v=List(),t);forprime(p=2, sqrtint(lim\4), forprime(q=p+1,sqrt(lim-p^2), if(isprime(t=p^2+3*p*q+q^2), listput(v,t), if(t>lim,break)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Nov 05 2012

is_A218771(n,v=0)={ my(r,c=0); isprime(n) & forprime( q=1,sqrtint(n\5), issquare(4*n+5*q^2, &r) || next; isprime((r-3*q)/2) || next; v || return(1); v>1 & print1([q,(r-3*q)/2]","); c++);c}  \\ - M. F. Hasler, Nov 05 2012

A219026 Number of primes p<=n such that 2n-p and 2n+p-2 are both prime.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 0, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 3, 3, 3, 3, 3, 1, 3, 1, 3, 2, 2, 2, 3, 4, 1, 4, 2, 2, 5, 1, 2, 4, 0, 3, 2, 3, 3, 2, 2, 1, 3, 3, 4, 4, 2, 3, 5, 3, 4, 3, 2, 3, 4, 3, 1, 3, 4, 3, 4, 3, 5, 4, 3, 3, 2, 3, 3, 3, 4, 2, 6, 3, 2, 7, 4, 2

Offset: 1

Zhi-Wei Sun, Nov 10 2012

nonn

Conjecture: a(n)>0 except for n=1,2,4,6,10,22,57.
This is stronger than the Goldbach conjecture; it has been verified for n up to 5*10^7.
Zhi-Wei Sun also conjectured that if n is not among 1,2,3,5,8,87,108 then there is a prime p in (n,2n)
such that 2n-p and 2n+p-2 are both prime. For conjectures in Section 2 of arXiv:1211.1588, he had similar conjectures with p<=n replaced by p in (n,2n)
For example, if n is not among 1,2,4,6,10,15 then there is a prime p in (n,2n) such that
2n-p and 2n+p+2 are both prime.

			a(8)=2 since 3 and 5 are the only primes p<=8 with 16-p and 14+p both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588v2.

Cf. A000040, A002375, A218754, A218585, A218654, A218656, A218825, A219023, A219025.

a[n_]:=a[n]=Sum[If[PrimeQ[2n-Prime[k]]==True&&PrimeQ[2n+Prime[k]-2]==True,1,0],{k,1,PrimePi[n]}]
Do[Print[n," ",a[n]],{n,1,20000}]
np[n_]:=Count[Prime[Range[PrimePi[n]]],?(AllTrue[{2n-#,2n+#-2},PrimeQ]&)]; Array[np,100] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Sep 23 2017 *)

A218797 Number of ways to write 2n - 1 as p + q + r with p <= q <= r and p, q, r, p^2 + q^2 + r^2 all prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 0, 1, 2, 2, 1, 3, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 4, 1, 3, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 2, 4, 4, 3, 0, 2, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 3, 3, 2, 3, 4, 2, 2, 3, 2, 1, 3, 3, 1, 2, 2, 5, 1, 4, 2, 2, 1, 1, 6, 3, 1, 5, 1, 1, 5, 4, 1, 4, 1, 2, 6, 2, 4, 2, 2, 2, 1, 4, 4

Offset: 1

Zhi-Wei Sun, Nov 05 2012

nonn

Conjecture: a(n) > 0 for all n=1715,1716,....

This conjecture is stronger than the weak Goldbach conjecture. It has been verified for n up to 500,000. Those 0

			a(7)=2 since 13=3+3+7=3+5+5, and both 3^2+3^2+7^2=67 and 3^2+5^2+5^2=59 are primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A054860, A085317, A218754, A218585, A218654, A218656.

a[n_]:=a[n]=Sum[If[PrimeQ[n-Prime[j]-Prime[k]]==True&&PrimeQ[Prime[j]^2+Prime[k]^2+(n-Prime[j]-Prime[k])^2]==True,1,0],{j,1,PrimePi[n/3]},{k,j,PrimePi[(n-Prime[j])/2]}]
Do[Print[n," ",a[2n-1]],{n,1,10000}]

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A219055 Number of ways to write n = p+q(3-(-1)^n)/2 with p>q and p, q, p-6, q+6 all prime.

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A218754 Number of ways to write n=p+q(3+(-1)^n)/2 with q<=n/2 and p, q, p^2+3pq+q^2 all prime.

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A218654 Number of ways to write n as x+y with 0

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A218825 Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.

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A218656 Number of ways to write 2n+1 as x+y with 0 < x < y and x^4 + y^4 prime.

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A219052 Number of ways to write n = p + q(3 - (-1)^n)/2 with q <= n/2 and p, q, p^2 + q^2 - 1 all prime.

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A219782 Number of ways to write n=x+y (0

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A218771 Primes of the form p^2 + 3pq + q^2 with p and q prime.

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